We first mention Lemma 4.3 page 59 in [1]} (and its corolary) : if \(m\ge 0\) , if \(\alpha >0\) , if \(\int _{K_\rho } \vert v \vert \le C \rho ^{m+\alpha }\)  then \(\int _{K_\rho } \vert x-y \vert ^{-m-\alpha /2} \vert v(x)\vert \, dx \le C_1(\alpha ,m,C,\mbox{diam}(\Omega ))\rho ^{\alpha /2}\)  for any \(y \in K_\rho \) .
We have denoted by \(\mbox{diam}(\Omega )\)  the diameter of \(\Omega \) .
Set \(v=\int _0^t \vert \nabla u_i \vert ^2\) .
Set \(m=N-2\)  and \(\alpha =(s-N)/s\) .
Assume \(s>N\)  so that \(\alpha >0\) .
We have \(m+\alpha =((N-1)s-N)/s\) .
We thus infer from (REF ) and Fubini's theorem that
\(\int _0^t \int _{K_\rho } \vert x-y \vert ^{-N+2-\alpha /2} \vert \nabla u_i \vert ^2&&= \int _{K_\rho } \vert x-y \vert ^{-N+2-\alpha /2}\int _0^t \vert \nabla u_i \vert ^2\nonumber \\&&\le C(C_s,s,N,t,\Vert u_i^0\Vert _{L^2(K_{2\rho })},\Vert Q \Vert _{L^{s/(s-1)}(K_{2\rho } \times (0,t))}) \rho ^{\alpha /2}\) 
